If \(f\)  is a trigonometric polynomial, say, \(\displaystyle f(t) = \sum _{k=0}^n a_k e^{i\lambda _k t}\) 
where \(a_k \in \mathbb {X}-\lbrace 0\rbrace \)  and \(\lambda _k \in \mathbb {R}\)  for \(k = 1, 2, ..., n\) , then \(\sigma _b(f) = \lbrace \lambda _k: \ k =1, 2, ..., n\rbrace \) . Moreover,
\(\frac{1}{\mu (Q_T)} \int _{Q_T} f(t) \nu (t) dt &=& a_0 \frac{\nu (Q_T)}{\mu (Q_T)} + \frac{1}{\mu (Q_T)} \int _{Q_T} \Big [\sum _{k=1}^n a_k e^{i\lambda _k t}\Big ] \nu (t) dt\\&=&a_0 \frac{\nu (Q_T)}{\mu (Q_T)} + \sum _{k=1}^n a_k \Big [\frac{1}{\mu (Q_T)}\int _{Q_T} e^{i\lambda _k t} \nu (t) dt\Big ]\\\) 
and hence
\( \left\Vert \frac{1}{\mu (Q_T)} \int _{Q_T} f(t) \nu (t) dt - a_0 \frac{\nu (Q_T)}{\mu (Q_T)}\right\Vert  &\le & \sum _{k=1}^n \left\Vert a_k\right\Vert  \Big |\frac{1}{\mu (Q_T)} \int _{Q_T} e^{i\lambda _k t} \nu (t) dt\Big |\\\) 
which by Eq. (REF ) yields
\(\left\Vert \frac{1}{\mu (Q_T)} \int _{Q_T} f(t) \nu (t) dt - a_0 \theta _{\mu \nu }\right\Vert  \rightarrow 0 \ \ \mbox{as} \ \ T \rightarrow \infty \) 
and therefore \({\mathcal {M}}(f, \mu , \nu ) = a_0 \theta _{\mu \nu } = \theta _{\mu \nu } M(f)\) .
If in the finite sequence of \(\lambda _k\)  there exist \(\lambda _{n_k} = 0\)  for \(k = 1, 2, ...l\)  with \(a_m \in \mathbb {X}-\lbrace 0\rbrace \)  for all \(m \ne  n_k\)  (\(k=1,2,...,l\) ), it can be easily shown that
\(\displaystyle {\mathcal {M}}(f, \mu , \nu ) = \theta _{\mu \nu } \sum _{k=1}^l a_{n_k} = \theta _{\mu \nu } M(f).\) 
Now if \(f: \mathbb {R}\mapsto \mathbb {X}\)  is an arbitrary almost periodic function, then for every \(\varepsilon > 0\)  there exists a trigonometric polynomial (Theorem REF ) \(P_\varepsilon \)  defined by
\(P_\varepsilon (t) = \sum _{k=1}^n a_k e^{i\lambda _k t}\) 
where \(a_k \in \mathbb {X}\)  and \(\lambda _k \in \sigma _b (f)\)  such that
\( \left\Vert f(t) - P_\varepsilon (t)\right\Vert  < \varepsilon \) 
for all \(t \in \mathbb {R}\) .
Proceeding as in Bohr [1]} it follows that there exists \(T_0\)  such that for all \(T_1, T_2 > T_0\) ,
\(\Big \Vert \frac{1}{\mu (Q_{T_1})} \int _{Q_{T_1}} P_\varepsilon (t) \nu (t) dt - \frac{1}{\mu (Q_{T_2})} \int _{Q_{T_2}} P_\varepsilon (t) \nu (t) dt \Big \Vert  = \theta _{\mu \nu } \Big \Vert M(P_\varepsilon ) - M(P_\varepsilon )\Big \Vert  = 0 < \varepsilon .\) 
In view of the above it follows that for all \(T_1, T_2 > T_0\) ,
\(\Big \Vert \frac{1}{\mu (Q_{T_1})} \int _{Q_{T_1}} f(t) \nu (t) dt - \frac{1}{\mu (Q_{T_2})} \int _{Q_{T_2}} f(t) \nu (t) dt \Big \Vert  &\le & \frac{1}{\mu (Q_{T_1})} \int _{Q_{T_1}} \Vert  f(t) - P_\varepsilon (t)\Vert  \nu (t) dt\\+ \Big \Vert \frac{1}{\mu (Q_{T_1})} \int _{Q_{T_1}} P_\varepsilon (t) \nu (t) dt - \frac{1}{\mu (Q_{T_2})} \int _{Q_{T_2}} P_\varepsilon (t) \nu (t) dt \Big \Vert \\+ \frac{1}{\mu (Q_{T_2})} \int _{Q_{T_2}} \Vert  f(t) - P_\varepsilon (t)\Vert  \nu (t) dt<3\varepsilon .\) 
